Some numerical and experimental advances in chaotic scattering - PowerPoint PPT Presentation

Some numerical and experimental advances in chaotic scattering Microlocal Analysis and Spectral Theory 2013 Maciej Zworski UC Berkeley September 28, 2013

A scattering problem 10 9 8 7 3 6 2 5 4 1 3 0 2 ! 1 1 ! 2 0 ! 3 ! 3 ! 2.5 ! 2 ! 1.5 ! 1 ! 0.5 0 0.5 1 1.5 3 a j e −| x − x j | 2 / b j � V ( x ) = j =1 We consider ih ∂ t u = − h 2 ∆ u + V ( x ) u � �

ih ∂ t u = − h 2 ∆ u + V ( x ) u

Newtonian dynamics: x ′ ( t ) = 2 ξ ( t ) , ξ ′ ( t ) = −∇ V ( x ( t )) , ϕ t ( x (0) , ξ (0)) := ( x ( t ) , ξ ( t )) . Trapped set at energy E : K E := { ( x , ξ ) : ξ 2 + V ( x ) = E , ϕ t ( x , ξ ) �→ ∞ , t → ±∞} . 10 9 8 7 3 6 2 5 4 1 3 0 2 ! 1 1 ! 2 0 ! 3 ! 3 ! 2.5 ! 2 ! 1.5 ! 1 ! 0.5 0 0.5 1 1.5

In the movies we saw the effects of Newtonian (classical) dynamics but we also saw oscillations, concentration and decay of waves. Quantum Resonances describe these waves resonating in interaction regions: there exist complex numbers z j ( h ) = E j ( h ) − i Γ j ( h ) , Γ j ( h ) > 0 , and w j ( x ) �∈ L 2 (resonant states), such that ( P − z j ( h )) w j = 0 , w j is outgoing .

Quantum Resonances describe the resonating waves: Potential 100 50 0 ! 3 ! 2 ! 1 0 1 2 3 4 5 Pole locations 0 ! 0.2 ! 0.4 ! 0.6 ! 0.8 ! 20 ! 15 ! 10 ! 5 0 5 10 15 20 Computed using squarepot.m http://www.cims.nyu.edu/ ∼ dbindel/resonant1d/

Here is how they sound: time = linspace(0,500,5000); sound(real(exp(-i*z*time)))

A real experimental example Potzuweit–Weich–Barkhofen–Kuhl–St¨ ockmann–Z ’12

Resonances for three discs: Barkhofen–Kuhl–Weich ’13

Resonances for three discs: O 3 s 2 0 0 s 1 s 3 O 1 0 O 2 ϕ ξ x ν ν 2 1 Barkhofen–Kuhl–Weich ’13

incoming set trapped set outgoing set Poon–Campos–Ott–Grebogi ’96

Resonances for three discs: Resonant states are microlocalized on the outgoing set: Helffer–Sj¨ ostrand ’85, Bony–Michel ’04, Nonnenmacher–Rubin ’07.

Sj¨ ostrand ’90: Suppose P = − h 2 ∆ + V where V is analytic (and reasonable). Suppose that the classical flow is hyperbolic on K E . Then resonances of of P , z j ( h ), satisfy # { z j ( h ) ∈ [ E − ǫ, E + ǫ ] − i [0 , h ] } ≤ Ch − m / 2 , m > dim ∪ | E ′ − E | < 2 ǫ K E ′ . Here the dimension is the Minkowski/box dimension: for M ⊂ R k , ǫ − γ vol R k ( { ρ : d ( ρ, M ) < ǫ } ) < ∞} . codim M = sup { γ : lim sup ǫ → 0 Earlier, non-geometric bounds: Regge ’58, Melrose ’82, Intissar ’86, Z ’87,’89.

Sj¨ ostrand ’90: # { z j ( h ) ∈ [ E − ǫ, E + ǫ ] − i [0 , h ] } ≤ Ch − m / 2 , m > dim ∪ | E ′ − E | < 2 ǫ K E ′ . K E := { ( x , ξ ) : ξ 2 + V ( x ) = E , ϕ t ( x , ξ ) �→ ∞ , t → ±∞} . ǫ − γ vol R k ( { ρ : d ( ρ, M ) < ǫ } ) < ∞} . codim M = sup { γ : lim sup ǫ → 0 E ! " " 10 9 8 + 7 E 6 3 " 5 2 t 4 1 ! ( " ) 3 0 2 ! 1 1 # ! 2 0 ! 3 ! 3 ! 2.5 ! 2 ! 1.5 ! 1 ! 0.5 0 0.5 1 1.5

More recently: Sj¨ ostrand–Z ’07: Resonances for − h 2 ∆ + V where V ∈ C ∞ c ( R n ; R ) (and more general operators) # { z j ( h ) ∈ [ E − h , E + h ] − i [0 , h ] } ≤ Ch − µ , 2 µ + 1 > dim K E . Nonnenmacher–Sj¨ ostrand–Z ’13: Resonances for − ∆ on R n \ � J j =1 O j (and more general operators). E ! " " + E " t ! ( " ) #

Numerical studies: Lin ’02:

Lu–Sridhar–Z ’03: The reason for showing the paper is to indicate that to communicate an idea it helps to publish it in physics.

Dyatlov ’13 (math) , Dyatlov–Z ’13 (physics) Weyl law for quasi-normal modes/resonances for perturbations of Kerr-de Sitter metrics (rotating black holes).

Dyatlov ’13 (math), Dyatlov–Z ’13 (physics) Weyl law for quasi-normal modes/resonances for perturbations of Kerr-de Sitter metrics (rotating black holes). The trapped set as a changes from 0 to 1: Wunsch–Z ’11: The key property of this smooth trapped set is the r -normal hyperbolicity for any r . Hirsch–Pugh–Schub ’77: stable under small C r perturbations.

Dyatlov ’13 (math), Dyatlov–Z ’13 (physics) Weyl law for quasi-normal modes/resonances for perturbations of Kerr-de Sitter metrics (rotating black holes). When the transversal expansion rates satisfy ν max < 2 ν min (valid for 98% of rotation speeds of black holes) then λ 2 # { z j ∈ the blue box } = (2 π ) 2 vol ( ∪ E < 1 K E )(1 + o (1)) , Sj¨ ostrand–Z ’99: Asymptotics for resonances for convex obstacles satisfying a pinching condition (cubic bands).

Dyatlov ’13 (math), Dyatlov–Z ’13 (physics) Weyl law for quasi-normal modes/resonances for perturbations of Kerr-de Sitter metrics (rotating black holes). When the transversal expansion rates satisfy ν max < 2 ν min (valid for 98% speeds of rotation of the black hole) then λ 2 # { z j ∈ the blue box } = (2 π ) 2 vol ( ∪ E < 1 K E )(1 + o (1)) , Faure–Tsujii ’13: Similar asymptotics for the Policott–Ruelle resonances for contact Anosov flows.

Dyatlov ’13 (math), Dyatlov–Z ’13 (physics) Weyl law for quasi-normal modes/resonances for perturbations of Kerr-de Sitter metrics (rotating black holes). When the transversal expansion rates satisfy ν max < 2 ν min (valid for 98% speeds of rotation of the black hole) then λ 2 # { z j ∈ the blue box } = (2 π ) 2 vol ( ∪ E < 1 K E )(1 + o (1)) , Faure–Tsujii ’13: Similar asymptotics for the Policott–Ruelle resonances for contact Anosov flows.

A simpler model. Nonnenmacher–Z ’05, ’07’: quantized open Baker maps (Balazs–Voros ’89, Saraceno ’90) Classical relation: q ′ = 3 q , p ′ = p / 3 , � 0 ≤ q ≤ 1 / 3 ( q , p ) ∼ ( q ′ , p ′ ) ⇐ ⇒ q ′ = 3 q − 2 , p ′ = ( p + 2) / 3 , 2 / 3 ≤ q < 1 . Quantum operator:  F N 0 0  M N = F ∗  . 0 0 0 3 N  0 0 F N ( F P is the discrete Fourier transform on C P ).

Open Baker map: incoming set trapped set outgoing set Three discs reduced to the boundary:

Open Baker map: Expected fractal Weyl law: for 0 < r < r 0 < 1,  F N 0 0  log 2 log 3 , M N = F ∗  . ♯ { λ ∈ Spec( M N ) , | λ | > r } ∼ N 0 0 0 3 N  0 0 F N

Nonnenmacher–Z ’07: for a simplified quantum Baker map corresponding to a complicated classical chaotic relation we have the fractal Weyl law for a sequence N = 3 k (the Walsh model).

Recent works in physics using variants of the quantum open maps (and other methods): Schomerus–Tworzyd� lo ’04, Keating et al ’06, Wiersig–Main ’08, Ramilowski et al ’09, Pedrosa et al ’09, Shepelyansky ’09, Shomerus–Wiersig–Main ’09, Ermann–Shepelyansky ’10, Kopp–Schomerus ’10, Ebersp¨ acher–Main–Wunner ’10, K¨ orber et al ’13. An interdisciplinary example:

A yet different setting: manifolds with hyperbolic ends Resonances defined as poles of ( − ∆ X − ( n − 1 − s ) s ) − 1 , continued from Im s > ( n − 1) / 2; X is a manifold with hyperbolic ends. Fractal upper bounds: Z ’99: Γ \ H 2 , Γ convex co-compact (based on Sj¨ ostrand ’90) e–Z ’04: Γ \ H 2 , Γ a Schottky group (based on some Lin–Guillop´ new Selberg zeta function techniques) Datchev–Dyatlov ’13: any manifold with hyperbolic ends (based on Sj¨ ostrand-Z ’07 and a new approach to meromorphic continuation by Vasy ’13) Other models using zeta functions: hyperbolic rational maps. Here the growth of zeros of the zeta function is related to the dimension of the Julia set. Strain-Z ’03, Christianson ’05.

Borthwick ’13:

Borthwick ’13: ℓ 1 = 10 , ℓ 2 = 12 , ϕ = 2 π/ 5

Borthwick ’13 Comparison with the fractal Weyl law:

Potzuweit–Weich–Barkhofen–Kuhl–St¨ ockmann–Z ’12 Experimental investigation of fractal Weyl laws.

Potzuweit–Weich–Barkhofen–Kuhl–St¨ ockmann–Z ’12 Experimental investigation of fractal Weyl laws. Left: The counting functions for R / a = 2 , 2 . 25 , 3 . 9 Fits of their slope for high frequencies are shown in blue. The orange curve over the lower histogram corresponds to the Weyl formula with 12% loss. Plotted in the inset is the difference between the Weyl formula with 12% loss and the experimental counting function for the closed system ( R / a = 2).

Potzuweit–Weich–Barkhofen–Kuhl–St¨ ockmann–Z ’12 Experimental investigation of fractal Weyl laws. Right: The data points correspond to the fitted exponent of the counting function in dependence of the R / a parameter. The three squares mark the examples which have already been presented in the previous figures. The darker shaded blue region indicates the R / a values without open channels; lighter shaded blue region has only a few open channels.

This may not seem to be so succesful but it lead to an interesting experiment about the gap between the real axis and resonances. Barkhofen–Weich–Potzuweit–Kuhl–St¨ ockmann–Z ’13 We look for γ > 0 such that there are no resonances in Im z > − γ, Re z > C 0